Mutually Exclusive Events

Mutually exclusive events are events that cannot happen at the same time. For example, if we toss a coin, the event of getting heads and the event of getting tails are mutually exclusive because both events cannot happen at the same time. Similarly, if we draw a card from a deck, the event of drawing a spade and the event of drawing a heart are mutually exclusive because a card cannot be both a spade and a heart at the same time.

Independence of Events

Two events are said to be independent if the occurrence of one event does not affect the probability of the occurrence of the other event. For example, if we toss a coin twice, the event of getting heads on the first toss and the event of getting heads on the second toss are independent because the outcome of the first toss does not affect the outcome of the second toss.

Relationship between Mutually Exclusive and Independent Events

If two events A and B are mutually exclusive, then they cannot happen at the same time. Therefore, the occurrence of one event affects the probability of the occurrence of the other event. This means that mutually exclusive events cannot be independent.

Equally Likely Events

Equally likely events are events that have the same probability of occurring. For example, if we toss a fair coin, the events of getting heads and getting tails are equally likely because they both have a probability of 1/2.

Mutually Exclusive Events cannot be Equally Likely

If two events A and B are mutually exclusive, then the probability of either event happening is the sum of their individual probabilities. For example, if we toss a coin, the probability of getting either heads or tails is 1/2 + 1/2 = 1. This means that mutually exclusive events cannot be equally likely because their individual probabilities add up to more than 1.

Conclusion

In conclusion, if two events A and B are mutually exclusive, then they cannot be independent because the occurrence of one event affects the probability of the occurrence of the other event. Similarly, mutually exclusive events cannot be equally likely because their individual probabilities add up to more than 1.